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Data-based Time Step Estimation

Data-based time step estimators for explicit time integration methods

Research project

Development and implementation of data-integrated methods for accurate estimation of the critical time step for time-explicit finite element simulations

Overview

Combination of classical numerical methods with machine learning for increased numerical efficiency and accuracy
Investigation of various machine learning methods for accurate estimation of the critical time step for explicit time integration methods
Extension of data-based time step estimation to complex finite elements and nonlinear simulations through innovative sampling strategies and consideration of additional influencing factors

Project description

Background

Explicit time integration methods are essential for simulating dynamic systems, such as in structural mechanics. The choice of time step significantly influences the stability, accuracy, and computation time of the simulation. Classical approaches often rely on simple heuristic or analytical estimates. Initial investigations show that these are neither accurate nor conservative for many element configurations. This can lead to inefficient or unstable simulations.

Project goals

The goal of this project is to develop data-based time step estimators that use machine learning algorithms to accurately and conservatively estimate the critical time step of explicit finite element simulations. This can save computing time, ensure the stability of the simulation, and improve the accuracy of the results.

Using two-dimensional quadrilateral elements, the great potential of this approach was demonstrated in Willmann et al. (2025): Data‐Based Estimation of Critical Time Steps for Explicit Time Integration.

Methodology

Data generation

All possible element configurations can be transformed into a parameter space in such a way that variables that have no influence on the critical time step or whose influence is easy to calculate are eliminated. The elements are rotated, translated, and scaled for this purpose. This results in 20 million element configurations representing all possible element shapes, see Fig. 1. For all configurations, the eigenvalue problem is solved to calculate the dimensionless critical time step as a reference solution.

Fig. 1: Various element configurations.

Evaluation of time step estimators and optimal safety factor functions

Half of the data set is used to evaluate time step estimators from literature and commercial software by comparing the estimated critical time step with the exact critical time step. This reveals how accurate these estimators are and how often they deliver non-conservative results. The results also allow the derivation of optimal safety factor functions for the various estimators, which ensure that all estimates are conservative and as accurate as possible.

New, data-based time step estimators

The second half of the data set can be used to train various machine learning approaches to estimate the critical time step for any element configuration. A new estimator based on a neural network (see Fig. 2) shows significantly higher accuracy than all existing approaches (see Fig. 3). Furthermore, the data-based estimators can be adapted to specific simulations in order to further increase their accuracy.

Fig. 2: Neural network for accurate prediction of the critical time step based on all possible element configurations (schematic illustration).

Fig. 3: Comparison of different estimators (all optimally conservative).

Central research questions

The project focuses on three central research questions that are crucial for the development of robust, data-based time step estimators:

Innovative sampling strategies and extension to complex finite elements

How can data-based time step estimation be transferred to more complex finite elements, such as volume or shell elements? Therefore, innovative sampling strategies are investigated that aim to cover the variety of possible element configurations as efficiently as possible. The goal is to make data-based time step estimation applicable to elements that are relevant in practice and more geometrically demanding.

Machine learning methods for time step estimation

Which machine learning methods—beyond neural networks—are particularly suitable for time step estimation? The focus is on generalizability, robustness, and efficient real-time evaluation. Approaches such as nonlinear regression, neural networks, and symbolic regression are evaluated in order to maximize prediction quality, minimize computational effort, and enable interpretability of the estimation.

Consideration of additional influencing factors on system level

Which parameters—especially in nonlinear simulations—must be included in the estimation in order to precisely determine the critical time step on system level? Here, we examine how geometric nonlinearities, current deformation states, or material parameters can be efficiently integrated. The goal is a conservative and accurate estimation that works reliably even with unstructured meshes and large deformations.

Publications

Willmann, T. (2025). 3D-shell element technology, nonlinear poisson stiffening and data-integrated time step estimation [Doktorarbeit, Bericht Nr. 80. Institut für Baustatik und Baudynamik der Universität Stuttgart]. https://doi.org/10.18419/opus-16843
- Abstract
Abstract
This thesis deals with three aspects of the finite element method: the development of three-dimensional shell elements, the explanation of a previously largely unknown stiffening effect, and a data-integrated approach for estimating the critical time step size in simulations with explicit time integration. A family of three-dimensional finite shell elements is developed by describing the velocity field as a polynomial in the out-of-plane coordinate while discretizing the shell plane with bilinear Lagrange polynomials. The elements use reduced integration in combination with an hourglass control. Numerical examples show that the developed shell elements with a quadratic or cubic velocity field in the out-of-plane coordinate yield more accurate results compared to elements with linear velocity field or elements based on the Reissner-Mindlin shell model. The thesis furthermore investigates and explains the nonlinear Poisson stiffening effect, a previously largely unknown stiffening effect. This effect is a nonlinear variant of Poisson thickness locking. Its causes and symptoms as well as mitigation strategies are discussed. Additionally, its relevance is demonstrated with a series of numerical examples. For explicit dynamic simulations, the thesis introduces a data-integrated approach to estimate the critical time step size. A time step estimator based on a neural network is developed for quadrilateral two-dimensional solid elements and shell elements with a linear velocity field in the out-of-plane coordinate. An update algorithm enhances computational efficiency, making this time step estimation method applicable in practical simulations.
Willmann, T., Schilling, M., & Bischoff, M. (2025). Data-Based Estimation of Critical Time Steps for Explicit Time Integration. International Journal for Numerical Methods in Engineering, 126, Article 4. https://doi.org/10.1002/nme.7666
- Abstract
Abstract
Finding the critical time step for conditionally stable time integration methods has been a decades-long problem. The apparently obvious option of directly computing it from a generalized eigenvalue analysis, identifying the largest eigenfrequency of the discrete system, is usually impractical because of its numerical expense and because a stiffness matrix is often unavailable in the context of explicit analysis. There exist two popular approaches to efficiently estimate the critical time step: A characteristic element length can be estimated based on heuristic formulas. The resulting estimate, however, cannot be guaranteed to be conservative. Another approach is to reformulate and simplify the underlying eigenvalue problem on the element level and to use certain inequalities to derive an upper bound for the largest eigenvalue. This is conservative but may show poor performance by significantly under-predicting the actual critical time step. Moreover, the necessary simplifications are usually specific to the investigated element formulation. Many works that develop time step estimators demonstrate their performance only for particular element configurations, making it difficult to compare the estimators. In this paper, data-driven approaches for time step estimation for 2d-elements that address several of the aforementioned problems are proposed. First, the set of all possible quadrilateral element geometries and its discrete representation are described. A detailed comparison of nine existing time step estimators based on more than ten million element configurations is presented. Additionally, the concept of an optimal safety factor function is introduced. This concept allows us to generate the optimal and conservative version of an existing estimator and thus solves two problems at the same time: It can be used to make non-conservative estimators conservative and to improve the performance of estimators that are conservative by construction. Finally, we formulate time step estimation as a function approximation problem. It allows us to derive customizable time step estimators solely based on data. Through two examples, we demonstrate that this data-driven approach yields time step estimators that outperform state-of-the-art estimators in terms of accuracy while also being efficient to evaluate.

Researcher:

Data-based time step estimators for explicit time integration methods

Overview